Degree reduction of Bézier cu

rve/surface

Lian Zhou

zhoulia5729@yahoo.com.cn

Dec. 14, 2006

Outline

Introduction of degree reduction in CAGD Related work Degree reduction of curves Degree reduction of tensor product Bézier surfaces Degree reduction of triangular Bézier surfaces Our work and future work

Problem Statement

Degree from to

Input: control points of

Output: control points of

Objective function:

di p R P

di q R Q

mind P,Q

n m

Applications

Data transfer and exchange Data compression Data comparison Surface intersection Curve smoothness Boolean operations and rendering

Michael S. Floater, High order approximation of rational curves by polynomial curves, Computer Aided Geometric Design 23 (2006) 621–628

CONSURF BUILD UNISURF CATIA COMPAC Geomod PADL GEMS

Early work

Based on the control points approaching Inverse of elevation

Forrest, A.R., Interactive interpolation and approximation by Bézier curve,

The Computer Journal, 15(1972), 71-79. G. Farin, Algorithms for rational Bezier curves, Computer Aided Design 15

(1983) 73–77.

Approximate conversion Danneberg, L., and Nowacki, H., Approximate conversion of surface represe

ntations with polynomial bases, Computer Aided Geometric Design, 2(198

5), 123-132. Hoschek, J., Approximation of spline curves, Computer Aided Geometric D

esign, 4(1987), 59-66.

Early work

Constrained optimization Moore, D. and Warren, J., Least-square approximation to Bezier curves and

surfaces in James Arvo eds. Computer Gemes (II), Academic Press, New Yo

rk, 1991. Lodha, S. and Warren, J., Degree reduction of Bezier simplexes, Computer

Aided Design, 26(1994), 735-746. Perturbing control points

胡事民， CAD 系统数据通讯中若干问题的研究 : [ 博士学位论文 ], 杭州 , 浙江大学数学系 , 1996.

Hu, S.M., Sun, J.G., Jin T.G., et al., Approximate degree reduction of Bezier curves, Tsinghua Science and Technology, 3(1998), 997-1000.

Early work

Based on the basis transformation Watkins, M. and Worsey, A., Degree reduction for Bézier curves, Computer Ai

ded Design, 20(1988), 398-405. Lachance, M.A., Chebyshev economization for parametric surfaces. Computer

Aided Geometric Design, 5(1988), 195-208. Eck, M., Degree reduction of Bézier curves, Computer Aided Geometric Desig

n, 10(1993), 237-257.69 Bogacki, P., Weinstein, S. and Xu, Y., Degree reduction of Bézier curves by un

iform approximation with endpoint interpolation, Computer Aided Design, 27(1995), 651-661.

Eck, M., Least squares degree reduction of Bézier curves, Computer Aided Design, 27(1995), 845-851.48

Recent work

Optimal multi-degree reduction Chen Guodong, Wang Guojin, Optimal multi-degree reduction of Bézier curves

with constraints of endpoints continuity. Computer Aided Geometric Design, 2002,19: 365-377

Zheng, J., Wang, G., Perturbing Bézier coefficients for best constrained degree reduction in the -norm. Graphical Models 2003, 65, 351–368.

Zhang Renjiang and Wang Guojin, Constrained Bézier curves’ best multi-degree reduction in the -norm, Progress in Natural Science, 2005, 15(9): 843-850

Others

2L

2L

Key progress

( , ) ( , ) ( , )00,0 0,1 0,

( , ) ( , ) ( , )1,1 1, 1, 1 I I I1

0 1 1 2 1

( , )( , ) ( , ),, , 1

1

2

1

, , , , , ,

nm n m n m nn m

m n m n m n nn m n m

r r r n s

m nm n m n nr n m rr r r r n r m

nr

nr

nn s

Bb b b

b b b B

bb b B

B

B

B

Q Q Q P P P

( , ) ( , ), ,

1 ( , )( , )1, 11, 1 1

( , ) ( , ) ( , ), , 1 ,

0 1 1 1

, , ,

, , , , , , , , ,

nm n m nm sm s m s m s n s

m s m m m nm n nm nm m n

m n m n m n nm m m n m n n

r r n s n s

Bb b

bb B

b b b B

Q Q Q

P P P P P P

0

1

,

n

n nnnn

B

B

B

P

Key progress

I 1 1 II 1 1 II( ) (1 ) ( ) (1 )s r s rn N N Nt t t t t t P P P B

II II 2 2 2 2 (2 2,2 2) III (2 2,2 2)( r , s ) r s r sN N N N N N N N

P B P E J P J

Jacobi

1, 1 III (2 2,2 2)1 r ssr s rMm Mt t t

P JJ

1

III (2 2,21 ) V

1

1 1 121 1s sr

m sr s m

M M M M ii

ri

r

t t t tt

P J P B Q B

B--J

Strength

Optimal Multi-degree reduction Explicit expression Precise error Less time consuming

Idea

Jacobi polynomial

Basis transformation

Key progress

Jacobi polynomial

.

.

.

mn

mndxxJxJxx rs

n

rsm

rsn

rs

,

,011 ,

,1

1

,

, , , , , ,1 1 0 0

s r s r s r s r s r s rn n n n nx b J x b J x b J x Q

n

k

krs

n

x

k

krsn

kn

snxJ

0

),(

2

1)(

1,1 , 1, 1 .x s r

Others

Lutterkort, D., Peters, J., Reif, U., 1999. Polynomial degree reduction in the -norm equals best Euclidean approximation of Bézier coefficients. Computer Aided Geometric Design 16, 607–612.

Ahn, Y.J., Lee, B.G., Park, Y., Yoo, J., 2004. Constrained polynomial degree reduction in the -norm equals best weighted Euclidean approximation of Bézier coefficients. Computer Aided Geometric Design 21, 181–191.

2L

2L

Optimal multi-degree reduction of Bézier

curves with -continuity

Lizheng Lu , Guozhao Wang

Computer Aided Geometric Design 23 (2006) 673–683

2G

Problem statement

Motivation

condition1G

Algorithm for -constrained degree reduction

1G

Least square method

Influence of the parameters , 0 1

A weakness

The approximation curve will be singular at the endpoint when or is nearly equal to 0.0v v

Regularization

Conjugate gradient method

Algorithm 1

Improvement of the singularities

Remark

-constrained degree reduction2G

Example 1

Example 2

Example 3

Degree reduction of tensor product Bézier surfaces

1 1

0 0 0 0

0 0 , 0 0

, , 0 0

, ,

0 ,0 ; 0,1; 0,1.

nm n m

u v u u v v

u v u vu v u v

o o u v

P Q

Related work 陈发来 , 丁友东 , 矩形域上参数曲面的插值降阶逼近 , 高等学校计算数学学

报 ( 计算几何专辑 ),1993,7,22-32 Hu Shimin, Zheng Guoqin, Sun Jiaguang. Approximate degree reduction of recta

ngular Bézier surfaces, Journal of Software, 1997, 4(4): 353-361 周登文 , 刘芳 , 居涛 , 孙家广 , 张量积 Bézier 曲面降阶逼近的新方法 , 计

算机辅助设计与图形学学报 , 2002 14(6), 553-556 Chen Guodong and Wang Guojin, Multi-degree reduction of tensor product Bézi

er surfaces with conditions of corners interpolations, SCIENCE IN CHINA, Series F,2002, 45(1): 51~58

郭清伟 , 朱功勤 , 张量积 Bézier 曲面降多阶逼近的方法 , 计算机辅助设计与图形学学报 , 2004,16(6)

章仁江 , CAGD 中曲线曲面的降阶与离散技术的理论研究 : [ 博士学位论文 ], 杭州 , 浙江大学数学系， 2004.

1 1

1 1 1 1

1 1

1 1

1 1 1 1 1 1

1 1

1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1

0 0 ,1 1

1 1

0 0 , 00 0 01 1

0, 0 ,0 0 ,

( , ) ( ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

m mm n m n

nm j j n j j nj j

n nn m n m n m

i i i m j mi i

n m n m n mm m n n n m n m

u v B v B u B v B u

B u B v B u B v B u B v

B u B v B u B v B u B

P Q Q

Q Q Q

Q Q Q

1

1

1 1

1 1 1 1

III III III00 01 0,

III IIITT 10 1,

III III III,0 ,1 ,

T M

( ))

( ) ( ) ( ) ( )

( ) ( )

0

m

mn n m m n m

n n n m

n n m m

v

u v u v

u v

P P P

P PB B B B

P P P

B B

P

P

mm

mnnn

m

m

nnmnmnijmn

1

1111

1

1

1

III,

III1,

III0,

III,1

III10

III,0

III01

III00

T

)1()1(

MM 0 E EPP

PPP

PP

PPP

P

－

1

1

2

0

2

0

22M1

1

M )()(~

)1()1()()(n

i

n

i

m

j

mj

niij

m

j

mj

niij vBuBvvuuvBuB PP

Our work

Multi-degree reduction of tensor

product Bézier surfaces in norm

Without constraintsWith interpolation

at endpoints

With boundary

constraints

Best Best locally Better

2L

Fruit 1

Control points

Approximate error

1 1 1 1 1 1

1 1

1

1

, , , , , , , , , ,

, , , , .

T T Tn n U m m U n n m m U n n U m mUR DL

Tn n

n

TmU m m UDR

Q B P B G B H H B G

H B H

F

F

1 1 1 1 1 1

1 1 1 1 1 1

, , , ,1 1 1 1 1 1 1 1

1

2

T T Tn m m n n m n n m mUR DL DRn m m n n m n n m mI Z B Z Z B Z Z B Z

Fruit 2

Control points are

Error bound is

1 1

1 11 1

1 1 1 11 1

,00 0, 0,

, ,,0 ,11 1

,,0 1, ,

1 1

TI u c Im m

v c C v cn nn m

TI u c In m n m

n m

p P p

Q P Q P

p P p

1 1 1 1

2, , 0 1 0 1

1 1 1( , )= .

1 1 16u u v v

nm n m nm n m Ld E

n m

P Q P Q

Example 1

Guo 0.4121 Chen 0.3183 Zhou 0.0421

Given a degree 6 6 surface,we will present the 1 1 reduced surface with

corners 0 continuous.

Bézier

Original surface

Error surface

Guo 0.4121 Chen 0.3183

Zhou 0.0421

Example 2

Given a degree 6 5 surface,we will present the 2 1 reduced surface with

corners 1 continuous.

Bézier

0.Gu 54o 18 0C .he 67n 04 0Z .ho 26u 04

Original surface

Error surface

0.Gu 54o 18 0C .he 67n 04

0Z .ho 26u 04

Example 3

Given a degree 7 7 surface,we will present the 2 2 reduced surface with

corners 1 continuous.

Bézier

0.Gu 35o 46 0Z .ho 32u 02 0C .he 07n 22

Original surface

Error surface

0.Gu 35o 46 0C .he 07n 22

0Z .ho 32u 02

Key progress1 1 2 2

M M 2 2

1 1 0 0

( ) ( ) (1 ) (1 ) ( ) ( )n m n m

n m n mij i j ij i j

i j i j

B u B v u u v v B u B v

P P

2 2 2 2

1,1 1,1M 2 2 M

0 0 0 0

( ) ( ) (2 1) (2 1)n m n m

n mij i j ij i j

i j i j

B u B v J u J v

P P

1 1

1 1

1 1

1 1

1 1

2 21,1 1,1M

0 0

2 22 2M

0 0

1 1

1 1

(1 ) (1 ) (2 1) (2 1)

ˆ(1 ) (1 ) ( ) ( )

( ) ( )

n m

ij i ji j

n mn m

ij i ji j

n mn m

ij i ji j

u u v v J u J v

u u v v B u B v

B u B v

P

P

Q Jacobi

Degree reduction of triangular Bézier surfaces

Refer to the report of Lizheng Lu in the Ph.D student seminar on Sep. 13

Related work Hu SM, Zuo Z, Sun JG. Approximate degree reduction of triangular Bézier surface.

Tsinghua Science and Technology 1998;3(2):1001–4 Rababah A. degree reduction of triangular Bézier surfaces with common tangent

planes at vertices. International Journal of Computational Geometry & Applications 2005;15(5):477–90.

郭清伟 , 陶长虹 , 三角 Bézier 曲面的降多阶逼近 . 复旦学报 ( 自然科学版 ) 2006 Vol.45 No.2 P.270-276

Lizheng Lu, Guozhao Wang, Multi-degree reduction of triangular Bézier surfaces with boundary constraints. Computer-Aided Design 38 (2006) 1215–1223

2L

Future work

Optimal approximation in various norm

Geometry continuous

Reduce the degree of a Bézier surface composed of some small Bézier surface holistically

Thanks!Thanks!

A lemma

A lemma

Gerald Farin

Degree: Ph.DUniversity of Braunschweig, 1979

Biography:

Gerald Farin joined ASU in 1987. He has also worked at the University of Utah and spent four years in CAD/CAM development at Mercedes-Benz, Stuttgart, Germany. He has taught CAGD tutorials worldwide and has given more than 100 invited lectures worldwide.

Research:- Computer Aided Geometric Design

- NURBS

- Modeling 3D